Galilei Space-times and Newton-Cartan Gravity. Tekin Dereli

Transcription

1 Galilei Space-times and Newton-Cartan Gravity Tekin Dereli Department of Physics, Koç University, İstanbul and The Turkish Academy of Sciences 1st Erdal İnönü Conference on Group Theory in Physics Middle East Technical University, Ankara 31 October 2008

2 The 4-dimensional space-time description of Newtonian gravity goes back to 1920s to the work of E. Cartan [1] and K. Friedrichs [2]. In the absence of gravity, the geometry of the non-relativistic flat space-time can be given in terms of independent space and time metric tensors that are both degenerate and the corresponding unique metric-compatible, torsion-free linear connection whose curvature vanishes. The revival of interest in these theories started in 1960s with the work of Trautman [3, 4] who wrote down Newton s law of gravity in terms of the curvature tensor of a suitably chosen connection. At about the same times differential geometry of degenerate manifolds were also studied [5, 6] from a more mathematical point of view.

3 Galilei Structure Let M be a smooth 4-manifold, g be a (0, 2) tensor field and τ be a (nonsingular) 1-form on M. The triple (M, g, τ) is called a Galilei structure provided the following are satisfied: 1. g is symmetric and degenerate of rank There exists a vector field V 0 such that g(v, X) = 0 for any vector field X, and τ(v ) = 1. We will fix such a vector field V and consider it as part of the Galilei structure. g will be called the space metric. The tensor product τ τ on M is a (0, 2) tensor that is symmetric, degenerate of rank 1. It will be called the time metric.

4 The tensor field g ε = ετ τ + g (1) where ε is a non-zero real parameter, is symmetric and non-degenerate. ḡ determines a unique symmetric, non-degenerate tensor field h of type (2, 0). There exists a unique (2, 0) tensor field h that is symmetric, degenerate of rank 3 given by h = 1 ε V V + h ε (2) where in terms of local coordinate components of h and g h µν g νλ = δ µ λ V µ τ λ. (3) Remark: The same Galilei structure on M can be completely determined by any one of the pairs (g, τ), (h, V ), (g, V ) or (h, τ).

5 Newton-Cartan Connections Let (M, g, τ) be a Galilei structure and be a connection on (M, g). We will call a Newton-Cartan connection if 1. X τ = 0 for any vector field X on M. The non-metricity of is given by ( X g)(y, Z) = Q(X, Y, Z) (4) where Q is the (0, 3) non-metricity tensor while the torsion of is found from the Cartan structure equations where T is the (1, 2) torsion tensor. defined by X Y Y X [X, Y ] = T (X, Y ) (5) The curvature operator R X,Y Z of R X,Y Z = X Y Z Y X Z [X,Y ] Z (6) is a type-preserving tensor derivation. The Riemann curvature tensor R of is the (1, 3) tensor field determined by the relation where β is an arbitrary 1-form. R(X, Y, Z, β) = β(r X,Y Z) (7)

6 Main Theorem Let (M, g, τ) be a Galilei structure, T be a (1, 2) tensor field such that T (X, Y ) = T (Y, X) and Q be a (0,3) tensor field with the properties Q(X, Y, Z) = Q(X, Z, Y ) and Q(X, V, V ) = 0 for all vector fields X, Y, Z and time vector field V. Then there exists a unique Newton-Cartan connection with torsion T and non-metricity Q iff 1. dτ(x, Y ) = τ(t (X, Y )), 2.(L V g)(x, Y ) = Q(X, Y, V ) Q(Y, X, V )+Q(V, X, Y )+g(t (V, X), Y )+g(x, T (V, Y )) where d is the exterior derivative and L V vector field V given previously. Lie derivative with respect to the Remark: We will call (M, g, τ, ) where is a torsion-free Newton-Cartan connection with Q 0 a degenerate Weyl space-time.

7 In a local coordinate chart {x µ }, connection coefficients of a torsion-free Newton-Cartan connection with non-metricity Q are given by Γ λ µν = 1 2 hλκ ( µ g κν + nu g κµ κ g µν ) + V λ µ τ ν 1 2 hλκ (Q µνκ + Q νµκ Q κµν ). (8) Furthermore λ g µν = Q λµν, λ h µν = h µρ h νσ Q λρσ. (9) If and ˆ are two torsion-free Newton-Cartan connections then their difference is a tensor. It is well known that the torsion-free Newton-Cartan connections are in one-to-one correspondence with 2-forms ω on M [5]. That is to say, in a local coordinate system Γ λ µν ˆΓ λ µν = h λκ ω κ(µ τ ν). (10)

8 Suppose, in particular we start with the unique torsion-free, metric compatible (Q = 0) connection ˆ that is determined completely by the metrics g and τ in a local coordinate system as ˆΓ λ µν = 1 2 hλκ ( µ g κν + nu g κµ κ g µν ) + V λ µ τ ν. (11) Let us consider from now on a related connection determined in terms of an arbitrary 2-form ω. The resulting non-metricity tensor Q turns out to be Q λµν = V κ (ω κ(λ τ µ) τ ν + ω κ(λ τ ν) τ µ ) + ω λ(µ τ ν) (12) in a local coordinate system, where ω µν = ω νµ and the round parentheses denote symmetrization as usual: ω µ(ν τ λ) = 1 2 (ω µντ λ + ω µλ τ ν ). Then a straightforward calculation shows λ h µν = 0. (13)

9 Gravitational Field Equations The gravitational field equations are postulated to be Ric = 4πGρτ τ (14) where G is the universal gravitational coupling constant and ρ is the mass density. To cut down the number of independent degrees of freedom in a correct way, (14) must be supplemented by the Trautman conditions h µκ Rκρλ ν = hνκ R µ κλρ. (15) Together with the above field equations we also postulate the autoparallel equations of motion of a freely falling point mass ĊĊ = 0 (16) where C : [0, 1] M is a timelike curve parametrized in terms of the time variable t such that τ = dt.

10 To illustrate the physical content of the above postulates, let us consider an adapted locally inertial coordinate system {x 0 = t, x 1 = x, x 2 = y, x 3 = z} in terms of which g ij = δ ij g 00 = 0 h ij = δ ij h 00 = 0 τ i = 0 τ 0 = 1 V i = 0 V 0 = 1 (17) where i, j = 1, 2, 3. These coordinates are defined up to kinematical transformations t ±t + a, x j ±R(t) j k xk ± v j t + b j (18) where R j k describe SO(3) rotations. Then we calculate the following nonvanishing connection coefficients: If we let Γ k 00 = ω k0 Γ k i0 = Γk 0i = 1 2 ω ki. (19) ω = τ dφ, (20) then the Trautman conditions are identically satisfied and the remaining equations reduce to φ = 4πGρ, x = φ. (21)

11 Comments The Cartan-Friedrichs formalism provides a consistent way to determine the Newtonian limit of relativistic space-times [7]. It can be used to delineate the minimal coupling of Newtonian gravitational field to the Schrödinger equation [8, 9]. A non-relativistic spin-1/2 electron wave function satisfies the first order Levy-Leblond equation that iterates to the usual Schrödinger equation [10, 11]. It is possible to write the Levy-Leblond equation directly in a Galilean covariant way by constructing the Dirac operator on the degenerate spinor bundle over a Newton-Cartan space-time [12, 13]. Lévy-Leblond equations ( i σ iq c ( i σ iq c ) A ) A ξ i 2m (E V + qφ) η = 0 η i 2mξ = 0

12 References 1. E. Cartan, Sur les variétés a connexion affine et la théorie de la relativité generalisée, Ann. Ecole Norm Sup. 40 (1922) 326, ibid 41 (1924) 1 2. K. Friedrichs, Eine invariante Formulierung des Newtonschen Gravitationsgesetzes und des Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz, Math. Ann. 98 (1927) A. Trautman, Sur la theorié Newtonienne de la gravitation, Comp. R. Acad. Sci. Paris 257 (1963) P. Havas, Four-dimensional formulations of Newtonian mechanics and their relation to the special and the general theory of relativity, Rev. Mod. Phys. 36 (1964) H. D. Dombrowski, K. Horneffer, Die Differentialgeometrie des Galileischen Relativätsprinzips, Math. Zeitschr. 86 (1964) M. Crampin, On differentiable manifolds with degenerate metrics, Proc. Camb. Phil Soc. 64 (1968) 307

13 7. J. Ehlers, Über den Newtonschen Grenzwert der Einsteinschen Gravitationstheorie in Grundlagenprobleme der Modernen Physik, J. Nitsch, J. Pfarr, E. W. Stachow (Eds.) (Bibliographisches Institut, Mannheim, 1981) 8. K. Kucha r, Gravitation, geometry and nonrelativistic quantum theory, Phys. Rev. D22 (1980) C. Duval, H. P. Künzle, Minimal gravitational coupling in the Newtonian theory and the covariant Schrödinger equation, Gen. Rel. Grav. 16 (1984) J. -M. Lévy-Leblond, Nonrelativistic particles and wave equations, Comm. Math. Phys. 6 (1967) H. P. Künzle, C. Duval, Dirac field on Newtonian space-time, Ann. Inst. Henri Poincarè, (1984) M. Limoncu, Newton-Lévy-Leblond Denklemleri, Doktora Tezi (Türkçe), Anadolu Üniversitesi, Ekim T. Dereli, Ş. Koçak, M. Limoncu, Linear connections on light-like manifolds, Turk. J. Math. 32 (2008) 41